Lazard Lie group
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Definition
Quick definition
A group is termed a Lazard Lie group if its 3-local nilpotency class is finite and less than or equal to the group's powering threshold.
Explicit definition
A group is termed a class Lazard Lie group for some natural number if both the following hold:
No. | Shorthand for property | Explanation |
---|---|---|
1 | The powering threshold for is at least , i.e., is powered for the set of all primes less than or equal to . | is uniquely -divisible for all primes . In other words, if is a prime and , there is a unique value satisfying . |
2 | The 3-local nilpotency class of is at most . | For any three elements of , the subgroup of generated by these three elements is a nilpotent group of nilpotency class at most . |
Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by more groups) as we increase . Thus, a particular value of may work for a group but larger and smaller values may not.
A group is termed a Lazard Lie group if it is a class Lazard Lie group for some natural number .
A Lazard Lie group is a group that can participate on the group side of the Lazard correspondence. The Lie ring on the other side is its Lazard Lie ring.
Set of possible values for which a group is a class Lazard Lie group
A group is a Lazard Lie group if and only if its 3-local nilpotency class is less than or equal to its powering threshold. The set of permissible values for which the group is a class Lazard Lie group is the set of satisfying:
3-local nilpotency class powering threshold
p-group version
A p-group is termed a Lazard Lie group if its 3-local nilpotency class is at most . In other words, every subgroup of it generated by at most three elements has nilpotency class at most where is the prime associated with the group.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subgroup-closed group property | No | Lazard Lie property is not subgroup-closed | It is possible to have a Lazard Lie group and a subgroup of such that is not a Lazard Lie group in its own right. |
quotient-closed group property | No | Lazard Lie property is not quotient-closed | It is possible to have a Lazard Lie group and a normal subgroup of such that the quotient group is not a Lazard Lie group in its own right. |
finite direct product-closed group property | No | Lazard Lie property is not finite direct product-closed | It is possible to have Lazard Lie groups and such that the external direct product is not a Lazard Lie group. |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
abelian group | any two elements commute | Precisely the case (see also Lazard correspondence#Particular cases) | |FULL LIST, MORE INFO | |
Baer Lie group | uniquely 2-divisible and class at most two | Precisely the case (see also Lazard correspondence#Particular cases | Global Lazard Lie group|FULL LIST, MORE INFO | |
p-group of nilpotency class less than p | global nilpotency class puts an upper bound on the 3-local nilpotency class | Global Lazard Lie group|FULL LIST, MORE INFO | ||
rationally powered nilpotent group | nilpotent and uniquely divisible for all primes | Global Lazard Lie group|FULL LIST, MORE INFO |